Without loss of generality, we have taken assumptions to be able to create our construction. Here we list our assumptions and explain why they can be used without harming the generality of the construction itself.

\textit{We have to add the points in such a way that they mimimize the forbidden area}: We will add a point $x$ to $L_1$, then we shall keep adding points in such a way that they all end up being neighbors of the point $x$ in the resulting graph. It is clear that if we have more available space ate each stage, then we will be able to add as many points as possible, so we need to add them in a way that minimizes the forbidden area $F$.

\textit{Ignoring the deltas}: In practice, each time we add a point we should put it farther away from the first point (otherwise this would be the first HSP-neighbor) and outside of the forbiden zone created by the points already set on the construction. If we set our points a very small $\delta$ further away from $x$ and then the construction ends up being essentially the same as if we were considering the very small changes of the distances between the points, hence, we can assume that all the points are at the same distance from $x$.

\textit{Points lie on the unit sphere}: It is a direct consecuence of the past assumption that we can choose any distance we want, so we will assume that all the points will lie on the unit sphere.
